[Paper Review] Degree conditions for matchability in $3$-partite hypergraphs

Ron Aharoni, Eli Berger+2 more|arXiv (Cornell University)|May 18, 2016

Limits and Structures in Graph Theory15 references3 citations

TL;DR

This paper establishes topological proofs for degree conditions ensuring large matchings in 3-partite hypergraphs, proving that 2n−1 matchings of size n in a bipartite graph admit a rainbow matching of size n under mild size constraints. It strengthens Drisko's theorem and extends results on Latin squares and transversals using combinatorial topology and degree-based conditions.

ABSTRACT

We study conjectures relating degree conditions in $3$-partite hypergraphs to the matching number of the hypergraph, and use topological methods to prove special cases. In particular, we prove a strong version of a theorem of Drisko \cite{drisko} (as generalized by the first two authors \cite{ab}), that every family of $2n-1$ matchings of size $n$ in a bipartite graph has a partial rainbow matching of size $n$. We show that milder restrictions on the sizes of the matchings suffice. Another result that is strengthened is a theorem of Cameron and Wanless \cite{CamWan}, that every Latin square has a diagonal (permutation submatrix) in which no symbol appears more than twice. We show that the same is true under the weaker condition that the square is row-Latin.

Motivation & Objective

• To prove that 2n−1 matchings of size n in a bipartite graph admit a rainbow matching of size n under relaxed size constraints.
• To generalize Drisko's theorem by weakening the requirement that all matchings have size exactly n.
• To extend results on Latin squares and transversals by proving that row-Latin squares admit partial transversals of size n−1 with no symbol repeated more than twice.
• To establish a topological proof for a strong version of the rainbow matching theorem with variable-sized matchings.
• To propose and investigate generalized degree conditions for matching number in 3-partite hypergraphs, including asymmetric and regularity-based conjectures.

Proposed method

• Employ topological methods, particularly the colorful Helly theorem and game-theoretic arguments, to prove the existence of large rainbow matchings.
• Use a game-theoretic model where players (POS and NON) alternately act on vertices to simulate matching construction and derive contradiction from score assumptions.
• Apply degree and simplicity constraints: (A,C) is simple, (B,C) is 2-simple, and deg(a) = n for all a ∈ A, to bound edge counts and derive contradictions.
• Introduce the concept of 'accommodating' sequences of minimum matching sizes and prove that a sequence is accommodating iff ai ≥ min(i,n) for all i ≤ 2n−1.
• Construct a double-copy hypergraph H′ from H to preserve simplicity and 2-simplicity, then apply Theorem 1.10 to deduce ν(H′) = n and infer ν(H) = n.
• Use contradiction arguments based on edge count bounds: assume ν(H) < n, derive upper bounds on |E(GK)|, and contradict a lower bound from degree conditions.

Experimental results

Research questions

• RQ1 Can the condition that all 2n−1 matchings have size exactly n in Drisko’s theorem be relaxed to allow smaller matchings?
• RQ2 Under what degree and simplicity conditions in 3-partite hypergraphs does the matching number ν(H) reach n?
• RQ3 Can the Brualdi-Stein conjecture on Latin squares be extended to row-Latin squares with no symbol repeated more than twice in a transversal?
• RQ4 Is the bound of 2n−1 matchings in Drisko’s theorem tight, and are there unique extremal configurations?
• RQ5 What are the minimal degree and simplicity conditions on a 3-partite hypergraph ensuring ν(H) ≥ n?

Key findings

• A family of 2n−1 matchings in a bipartite graph, where |Fi| ≥ i for i ≤ n−1 and |Fi| = n for i ≥ n, admits a rainbow matching of size n.
• The condition that |Fi| ≥ min(i,n) for all i ≤ 2n−1 is both necessary and sufficient for the existence of a rainbow matching of size n.
• The hypergraph version of Drisko’s theorem holds under weaker conditions: if |A| ≥ 2n−1, deg(a) = n for all a ∈ A, (A,C) is simple, and (B,C) is 2-simple, then ν(H) ≥ n.
• The conjecture that every row-Latin square has a partial transversal of size n−1 with no symbol repeated more than twice is confirmed.
• The construction of a double-copy hypergraph H′ preserves simplicity and 2-simplicity, enabling the deduction of ν(H) = n from ν(H′) = n.
• A contradiction is derived by assuming ν(H) < n, leading to an upper bound on |E(GK)| that violates a lower bound from degree and simplicity constraints.